The present invention relates generally to methods for tracking magnetic source objects, and more particularly, to a linear perturbation method that provides for Kalman filter tracking of magnetic field sources.
The assignee of the present invention has heretofore developed numerous inventions that describe methods for locating an idealized magnetic dipole moment source by processing a set of magnetic field measurements collected at an array of points that may be arbitrarily distributed in space and time (i.e. fixed or moving sensors in any combination or arrangement). According to Maxwell's equations, the magnetic field of a dipole moment is a nonlinear function of the field position so that the field equation cannot be solved in terms of the source location. However, if the magnitude and orientation of the source's dipole moment vector is known then a set of matched filter processes may be applied to the measurements to determine the most likely source location. In this approach the volume of space containing the dipole source is represented by a grid of hypothesized locations and one matched filter process is executed per grid node. The resulting response will be maximum at the node nearest the actual dipole thereby indicating the approximate source location.
In practice, however, the dipole moment vector of the source is generally unknown. When this is the case, its three vector components are allowed to vary over the space of probable values, which is also partitioned into a discrete grid with each node representing the direction and magnitude of the dipole moment. Now the matched filter response is maximized over both grids simultaneously to determine both the unknown source moment and its location. U.S. Pat. No. 5,731,996 assigned to the assignee of the present invention teaches that the unknown dipole moment may be linearly estimated at each source location and that the estimating process may be integrated into the matched filter, thereby reducing the unknown source problem to nearly the same level of processing difficulty as the known source problem.
Relevant prior art known to the present inventors is as follows. U.S. Pat. No. 5,239,474 entitled "Dipole Moment Detection and Localization" discloses an algorithm that localizes a magnetic dipole moment source by maximizing an objective function over a grid of search points that spans the search volume. The objective function used is the correlation between the Anderson function coefficients of the signal, normalized by a range factor, and the Anderson coefficients of a dipole source with known or hypothesized orientation. An alternative objective function briefly mentioned is the correlation between the signal, normalized by a range factor, and the ideal signal of a hypothesized dipole source and location. Both methods are dependent on the Anderson functions in spherical coordinates which are suitable for linear arrays of magnetic field sensors but are difficult to apply to more general array geometries.
U.S. Pat. No. 5,731,996 entitled "Dipole moment detector and localizer" discloses an algorithm that also localizes by maximizing an objective function over a search grid. However, the magnetic dipole moment source does not need to be known or hypothesized because it is uniquely estimated at each grid point. Also, the more general electromagnetic field moment equation in Cartesian coordinates is used in lieu of Anderson functions to model the dipole field. This makes the method more suitable for arbitrary array geometries. The objective function is the correlation between the signal and the ideal signal of the estimated dipole source.
U.S. Pat. No. 5,684,396 entitled "Localizing magnetic dipoles using spatial and temporal processing of magnetometer data" discloses an algorithm that performs the same basic operations as U.S. Pat. No. 5,731,996 except that a moving dipole source may be hypothesized in lieu of a fixed one. The magnetic field is measured over a period of time, as well as various points in space, and the source position and track during the time period are both hypothesized. The same dipole estimate is sued and the same objective function is maximized to determine the most likely position and track.
U.S. patent application Ser. No. 09/174,682, filed Oct. 19, 1998, entitled "Magnetic Object Tracking Based on Direct Observation of Magnetic Sensor Measurements" (having assignee docket number PD-970267) discloses an algorithm having an entirely new approach to solving the magnetic dipole moment localization problem for a source that is moving. It assumes that the source has been detected by other means and the initial position and velocity have also been determined. It then uses a Kalman filter to track the arbitrary movement of the source using only the magnetic field measurements collected over an array of sensors. The Kalman filter models the probable behavior of the source and the physical relationship between the moving source and its changing dipole field. The method offers two advancements. First, the computational load to localize the moving source at each point in time is greatly reduced. Second, the plant model greatly enhances the estimates of location based on the magnetic field measurements.
Other prior art documents are as follows. A report was published by Irving S Reed and is entitled Adaptive Space-Time Airborne Arrays for Magnetic Anomaly Detection (MAD) of Submarines, Final Report of DARPA Contract DAAB07-86-C-S009, Adaptive Sensors Incorporated, March 1987. This report describes a method of detecting and localizing submarines as magnetic dipole sources that includes some elements disclosed in U.S. Pat. No. 5,239,474 and U.S. Pat. No. 5,731,996 cited above. Spherical coordinates and Anderson functions are used as in item 1, while an Anderson coefficients estimator similar to the dipole moment estimator of item 2 is used to eliminate dependence on the unknown dipole moment. An objective function very similar to the function disclosed in U.S. Pat. No. 5,731,996, although based on Anderson functions, is used to localize the source. However, because of the ambiguity inherent in the array geometry and scalar field sensor used in the MAD application, localization is limited to two dimensions.
An article was published by Keiichi Mori, and is entitled "Application of Weight Functions to the Magnetic Localization of an Object", IEEE Transactions on Magnetics, May 1989, Vol. 25, No. 3, pp. 2726-2731. This article describes a method of localizing magnetic dipole sources that is similar to the method disclosed in U.S. Pat. No. 5,731,996 cited above. The principal difference is that the objective function is a weighted least squares cost function rather than the signal correlation. Two means of establishing the weights are described but neither is based in the statistics of the problem. Therefore, this prior art method leads to a different localization result than the method disclosed in U.S. Pat. No. 5.831,996, and the result can be shown to be both asymptotically biased and suboptimal in a statistical sense.
The principal disadvantage of each of these prior art methods is that localization must be performed by computing the objective function at each grid point in a search volume in order to find the maximum. The reason for this is that the objective function depends in a nonlinear way on the hypothesized source location. The computational cost goes up exponentially with the number of unknowns. If the search volume is divided into ten sectors along each axis then localization requires evaluating the objective function at 1,000 grid points. If the source is in motion then the problem rapidly becomes more complex. If a simple linear motion is assumed and the velocity vector is hypothesized at ten possible values along each axis then the objective function must be evaluated at 1,000,000 grid points. Curvilinear motion is even more difficult.
Localization may be posed as a classical nonlinear optimization problem and any of a wide variety of nonlinear solution algorithms may be used to extract a solution in the neighborhood of an initial point. These methods may reduce the amount of computation substantially. However, if the signal-to-noise ration (SNR) is low then a large number of initial points must be tested because of the presence of local maxima arising from the noise.
Accordingly, it is an objective of the present invention to provide for a linear perturbation method that provides for Kalman filter tracking of magnetic field sources.